Abstract:
Much of the theory of cardinal characteristics of the continuum is concerned with cardinal numbers of the form "the minimum cardinality of an adequate set of reals" for some interpretation of "adequate". For example, "adequate" could mean "cofinal in the pointwise ordering of sequences of natural numbers" or it could mean "not of Lebesgue measure zero"; many other meanings have also been studied. More recently, the theory has looked more closely at adequate sets themselves (in all these senses of "adequate") rather than only their cardinalities. As a result, attention has also been paid to modifications of these notions of adequacy, modifications that do not change the minimum cardinalities but do influence the combinatorial properties of these sets of reals. I plan to survey some of the background of the subject and then describe a few of these newer results.